{"title":"Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for\u0000 vector bundles","authors":"Y. Hashimoto, Julien Keller","doi":"10.46298/epiga.2022.6577","DOIUrl":"https://doi.org/10.46298/epiga.2022.6577","url":null,"abstract":"For a holomorphic vector bundle $E$ over a polarised K\"ahler manifold, we\u0000establish a direct link between the slope stability of $E$ and the asymptotic\u0000behaviour of Donaldson's functional, by defining the Quot-scheme limit of\u0000Fubini-Study metrics. In particular, we provide an explicit estimate which\u0000proves that Donaldson's functional is coercive on the set of Fubini-Study\u0000metrics if $E$ is slope stable, and give a new proof of Hermitian-Einstein\u0000metrics implying slope stability.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on Lang's conjecture for quotients of bounded domains","authors":"S. Boucksom, Simone Diverio","doi":"10.46298/epiga.2021.volume5.6050","DOIUrl":"https://doi.org/10.46298/epiga.2021.volume5.6050","url":null,"abstract":"It was conjectured by Lang that a complex projective manifold is Kobayashi\u0000hyperbolic if and only if it is of general type together with all of its\u0000subvarieties. We verify this conjecture for projective manifolds whose\u0000universal cover carries a bounded, strictly plurisubharmonic function. This\u0000includes in particular compact free quotients of bounded domains.\u0000\u0000 Comment: 10 pages, no figures, comments are welcome. v3: following suggestions\u0000 made by the referee, the exposition has been improved all along the paper, we\u0000 added a variant of Theorem A which includes manifolds whose universal cover\u0000 admits a bounded psh function which is strictly psh just at one point, and we\u0000 added a section of examples. Final version, to appear on 'Epijournal G'eom.\u0000 Alg'ebrique","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the B-Semiampleness Conjecture","authors":"E. Floris, Vladimir Lazi'c","doi":"10.46298/epiga.2019.volume3.5063","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.5063","url":null,"abstract":"The B-Semiampleness Conjecture of Prokhorov and Shokurov predicts that the\u0000moduli part in a canonical bundle formula is semiample on a birational\u0000modification. We prove that the restriction of the moduli part to any\u0000sufficiently high divisorial valuation is semiample, assuming the conjecture in\u0000lower dimensions.\u0000\u0000 Comment: 26 pages","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48063009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The equivalence of several conjectures on independence of $ell$","authors":"R. V. D. D. Bruyn","doi":"10.46298/epiga.2020.volume4.5570","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5570","url":null,"abstract":"We consider several conjectures on the independence of $ell$ of the 'etale\u0000cohomology of (singular, open) varieties over $bar{mathbf F}_p$. The main\u0000result is that independence of $ell$ of the Betti numbers\u0000$h^i_{text{c}}(X,mathbf Q_ell)$ for arbitrary varieties is equivalent to\u0000independence of $ell$ of homological equivalence $sim_{text{hom},ell}$ for\u0000cycles on smooth projective varieties. We give several other equivalent\u0000statements. As a surprising consequence, we prove that independence of $ell$\u0000of Betti numbers for smooth quasi-projective varieties implies the same result\u0000for arbitrary separated finite type $k$-schemes.\u0000\u0000 Comment: Published version. 27 pages","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48130581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Pluricomplex Green's functions and Fano manifolds","authors":"Nicholas McCleerey, Valentino Tosatti","doi":"10.46298/epiga.2019.volume3.4706","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.4706","url":null,"abstract":"We show that if a Fano manifold does not admit Kahler-Einstein metrics then\u0000the Kahler potentials along the continuity method subconverge to a function\u0000with analytic singularities along a subvariety which solves the homogeneous\u0000complex Monge-Ampere equation on its complement, confirming an expectation of\u0000Tian-Yau.\u0000\u0000 Comment: EpiGA Volume 3 (2019), Article Nr. 9","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Infinite families of inequivalent real circle actions on affine four-space","authors":"L. Moser-Jauslin","doi":"10.46298/epiga.2019.volume3.4685","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.4685","url":null,"abstract":"\u0000International audience\u0000\u0000 The main result of this article is to construct infinite families of non-equivalent equivariant real forms of linear C*-actions on affine four-space. We consider the real form of $mathbb{C}^*$ whose fixed point is a circle. In [F-MJ] one example of a non-linearizable circle action was constructed. Here, this result is generalized by developing a new approach which allows us to compare different real forms. The constructions of these forms are based on the structure of equivariant $mathrm{O}_2(mathbb{C})$-vector bundles.\u0000 Le résultat principal de cet article est de construire des familles infinies de formes réelles équivariantes, non équivalentes entre elles, d’actions linéaires de $mathbb{C}^*$ sur l’espace affine de dimension 4. L’article [F-MJ] construisait un exemple d’action du cercle non linéarisable. Ici nous généralisons ce résultat en développant une nouvelle approche qui nous permet de comparer les différentes formes réelles. Les constructions de ces formes réelles s’appuient sur la structure de $mathrm{O}_2(mathbb{C})$-fibrés vectoriels équivariants.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Remarks on the positivity of the cotangent bundle of a K3 surface","authors":"Frank Gounelas, J. C. Ottem","doi":"10.46298/epiga.2020.volume4.5924","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5924","url":null,"abstract":"Using recent results of Bayer-Macr`i, we compute in many cases the\u0000pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth\u0000projective K3 surface. We then use these results to construct explicit families\u0000of smooth curves on which the restriction of the cotangent bundle is not\u0000semistable (and hence not nef). In particular, this leads to a counterexample\u0000to a question of Campana-Peternell.\u0000\u0000 Comment: Published version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rigidity properties of holomorphic Legendrian singularities","authors":"Jun-Muk Hwang","doi":"10.46298/epiga.2019.volume3.4495","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.4495","url":null,"abstract":"We study the singularities of Legendrian subvarieties of contact manifolds in\u0000the complex-analytic category and prove two rigidity results. The first one is\u0000that Legendrian singularities with reduced tangent cones are\u0000contactomorphically biholomorphic to their tangent cones. This result is partly\u0000motivated by a problem on Fano contact manifolds. The second result is the\u0000deformation-rigidity of normal Legendrian singularities, meaning that any\u0000holomorphic family of normal Legendrian singularities is trivial, up to\u0000contactomorphic biholomorphisms of germs. Both results are proved by exploiting\u0000the relation between infinitesimal contactomorphisms and holomorphic sections\u0000of the natural line bundle on the contact manifold.\u0000\u0000 Comment: 21 pages, minor revision","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The parabolic exotic t-structure","authors":"Pramod N. Achar, Nicholas J Cooney, S. Riche","doi":"10.46298/epiga.2018.volume2.4520","DOIUrl":"https://doi.org/10.46298/epiga.2018.volume2.4520","url":null,"abstract":"International audience\u0000 \u0000 Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks.\u0000 \u0000 \u0000 Soit G un groupe algébrique réductif connexe sur un corps k algébriquement clos. La t-structure exotique sur le fibré cotangent de sa variété de drapeaux T^*(G/B), introduite à l'origine par Bezrukavnikov, a été un outil clé pour de nombreux résultats majeurs en théorie géométrique des représentations, en particulier la démonstration de la conjecture de Finkelberg-Mirkovic graduée. Dans cet article, nous étudions (sous de légères hypothèses techniques) une t-structure analogue sur le fibré cotangent de la variété de drapeaux partiels T^*(G/P). Comme application, nous prouvons un analogue parabolique de l'équivalence de Arkhipov-Bezrukavnikov-Ginzburg. Lorsque la caractéristique de k est supérieure au nombre de Coxeter, nous déduisons un analogue de la conjecture de Finkelberg-Mirkovic graduée pour certains blocs singuliers.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hamiltonian actions of unipotent groups on compact K\"ahler manifolds","authors":"D. Greb, C. Miebach","doi":"10.46298/epiga.2018.volume2.4486","DOIUrl":"https://doi.org/10.46298/epiga.2018.volume2.4486","url":null,"abstract":"We study meromorphic actions of unipotent complex Lie groups on compact\u0000K\"ahler manifolds using moment map techniques. We introduce natural stability\u0000conditions and show that sets of semistable points are Zariski-open and admit\u0000geometric quotients that carry compactifiable K\"ahler structures obtained by\u0000symplectic reduction. The relation of our complex-analytic theory to the work\u0000of Doran--Kirwan regarding the Geometric Invariant Theory of unipotent group\u0000actions on projective varieties is discussed in detail.\u0000\u0000 Comment: v2: 30 pages, final version as accepted by EPIGA","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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